Measuring Supplementary Influence by Using Sequential Linear Regression

The new measurement concept of ‘supplementary' influence is defined within the set-up of descriptive multiple regression analysis. The supplementary coefficient describes the average change of the regressand that can additionally be attributed to one regressor when several other regressors are already taken into account. In the case of a high degree of multicollinearity, this supplementary coefficient logically tends to be small. In order to motivate the definition, Frisch and Waugh's (Econometrica 1 (1933) 387) representation of the partial coefficient is reconsidered with respect to its descriptive properties. A simple modification yields the defining formula for the supplementary coefficient. The fact that this coefficient just is the product of the partial coefficient and the tolerance of the regressor can be shown. Further, its role in a decomposition of the slope coefficient of a univariate regression is demonstrated. Sequential regression, which uses a sequence of regression models, provides a way to define the supplementary coefficient independently from the concept of partial influence. In this article, all coefficients only depend on the covariance matrix of the regressand and the regressors. The regressand as well as all regressors are assumed to be square integrable random variables